a-3-x-3-x-2-a-1-x-7-0-is-a-cubic-polynomial-in-x-whose-Roots-are-positive-real-numbers-satisfying-225-2-2-7-144-2-2-7-100-2-2-7-find-a-1-

Question Number 195027 by York12 last updated on 22/Jul/23

a_{3} x^{3} - x^{2} + a_{1} x - 7 = 0 i s a c u b i c p o l y n o m i a l i n x

w h o s e R o o t s a r e α, β, γ p o s i t i v e r e a l n u m b e r s

s a t i s f y i n g

\frac{225 α^{2}}{α^{2} + 7} = \frac{144 β^{2}}{β^{2} + 7} = \frac{100 γ^{2}}{γ^{2} + 7}

f i n d (a_{1})

Commented by mr W last updated on 25/Jul/23

i g o t a_{1} = \frac{77}{15}

Commented by York12 last updated on 25/Jul/23

p l e a s e s i r e x p l a i n w h y

Answered by mr W last updated on 25/Jul/23

\Rightarrow α + β + γ = \frac{1}{a_{3}}

{(\frac{1}{x})}^{3} - \frac{a_{1}}{7} {(\frac{1}{x})}^{2} + \frac{1}{7} (\frac{1}{x}) - \frac{a_{3}}{7} = 0

\Rightarrow \frac{a_{1}}{7} = \frac{1}{α} + \frac{1}{β} + \frac{1}{γ}

a_{3} x^{3} - x^{2} + a_{1} x - 7 = 0

x^{2} + 7 = x (a_{3} x^{2} + a_{1})

\frac{225 α^{2}}{α^{2} + 7} = \frac{144 β^{2}}{β^{2} + 7} = \frac{100 γ^{2}}{γ^{2} + 7} = \frac{1}{k}, s a y

\frac{225 α^{2}}{α (a_{3} α^{2} + a_{1})} = \frac{144 β^{2}}{β (a_{3} β^{2} + a_{1})} = \frac{100 γ^{2}}{γ (a_{3} γ^{2} + a_{1})} = \frac{1}{k}

\Rightarrow a_{3} α + \frac{a_{1}}{α} = 225 k

\Rightarrow a_{3} β + \frac{a_{1}}{β} = 144 k

\Rightarrow a_{3} γ + \frac{a_{1}}{γ} = 100 k

\Rightarrow a_{3} (α + β + γ) + a_{1} (\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}) = 469 k

\Rightarrow 1 + \frac{a_{1}^{2}}{7} = 469 k

\Rightarrow a_{1} = \sqrt{7 (469 k - 1)}

\frac{225}{1 + \frac{7}{α^{2}}} = \frac{144}{1 + \frac{7}{β^{2}}} = \frac{100}{1 + \frac{7}{γ^{2}}} = \frac{1}{k}

\Rightarrow \frac{\sqrt{7}}{α} = \sqrt{225 k - 1}

\Rightarrow \frac{\sqrt{7}}{β} = \sqrt{144 k - 1}

\Rightarrow \frac{\sqrt{7}}{γ} = \sqrt{100 k - 1}

\sqrt{7} (\frac{1}{α} + \frac{1}{β} + \frac{1}{γ}) = \sqrt{225 k - 1} + \sqrt{144 k - 1} + \sqrt{100 k - 1}

\frac{a_{1}}{\sqrt{7}} = \sqrt{225 k - 1} + \sqrt{144 k - 1} + \sqrt{100 k - 1}

\Rightarrow \sqrt{469 k - 1} = \sqrt{225 k - 1} + \sqrt{144 k - 1} + \sqrt{100 k - 1}

\Rightarrow \sqrt{225 k - 1} + \sqrt{144 k - 1} = \sqrt{469 k - 1} - \sqrt{100 k - 1}

\Rightarrow \sqrt{(225 k - 1) (144 k - 1)} = 100 k - \sqrt{(469 k - 1) (100 k - 1)}

\Rightarrow 245 k - 2 = 2 \sqrt{(469 k - 1) (100 k - 1)}

\Rightarrow 127575 k = 1296

\Rightarrow k = \frac{1296}{127575} = \frac{16}{1575}

a_{1} = \sqrt{7 (\frac{469 \times 16}{1575} - 1)} = \frac{77}{15} ✓

α = \sqrt{\frac{7}{225 \times \frac{16}{1575} - 1}} = \frac{7}{3}

β = \sqrt{\frac{7}{144 \times \frac{16}{1575} - 1}} = \frac{35}{9}

γ = \sqrt{\frac{7}{100 \times \frac{16}{1575} - 1}} = 21

a_{3} = \frac{1}{\frac{7}{3} + \frac{35}{9} + 21} = \frac{9}{245}

Commented by York12 last updated on 25/Jul/23

t h a n k s s o m u c h

Facebook Tweet Pin